Enter either position uncertainty (σₓ) or momentum uncertainty (σₚ) — and this Heisenberg's Uncertainty Principle Calculator solves for the missing value using the relation σₓ·σₚ ≥ ℏ/2. Optionally provide the particle's mass to also get the minimum velocity uncertainty (σᵥ). Results are shown in standard SI units.
Calculated Uncertainty
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Result (Scientific Notation)
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Minimum Bound (ℏ/2)
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σₓ · σₚ Product
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Velocity Uncertainty (σᵥ)
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Heisenberg's Uncertainty Principle, formulated by Werner Heisenberg in 1927, states that it is fundamentally impossible to simultaneously know both the exact position and exact momentum of a quantum particle. The more precisely one property is measured, the less precisely the other can be known — this is not a limitation of instruments but an intrinsic feature of quantum mechanics.
The standard formulation is σₓ · σₚ ≥ ℏ/2, where σₓ is the standard deviation of position, σₚ is the standard deviation of momentum, and ℏ (h-bar) is the reduced Planck constant ≈ 1.0546 × 10⁻³⁴ J·s. The minimum product σₓ · σₚ can never be less than ℏ/2 ≈ 5.273 × 10⁻³⁵ J·s.
Use the relation σₚ ≥ ℏ / (2 · σₓ). Simply divide the reduced Planck constant ℏ by twice the known position uncertainty. The result gives the minimum possible standard deviation of the particle's momentum. This calculator performs that computation automatically.
A common misconception is that the act of observing disturbs the particle and causes the uncertainty. While measurement does interact with quantum systems, the uncertainty principle is more fundamental — it arises from the wave-like nature of quantum objects. Even a perfectly non-invasive measurement cannot violate the principle.
The reduced Planck constant ℏ ≈ 1.055 × 10⁻³⁴ J·s is extraordinarily small. For macroscopic objects (e.g., a baseball), the resulting uncertainties are so tiny they are completely undetectable. The principle becomes physically relevant only for atomic and sub-atomic particles such as electrons, protons, or photons.
Position uncertainty is entered in nanometres (nm) and internally converted to metres. Momentum uncertainty is displayed in units of 10⁻²⁷ kg·m/s for readability. Mass is entered in units of 10⁻³¹ kg (so the electron mass ≈ 9.109). All outputs are in standard SI units (J·s, m/s).
Yes — if you provide the particle's mass m, the calculator computes σᵥ = σₚ / m. For an electron (m ≈ 9.109 × 10⁻³¹ kg), even a position uncertainty of 1 nm leads to a velocity uncertainty of hundreds of thousands of metres per second, illustrating how dramatic quantum uncertainty is at small scales.
They are identical. ℏ (h-bar) is defined as h / (2π), so ℏ/2 = h / (4π). Both expressions represent the same minimum bound. Some textbooks write σₓ · σₚ ≥ h/(4π) while others use σₓ · σₚ ≥ ℏ/2 — the physics is exactly the same.